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In the following, we present some clues which show that replacing the Ricci curvature tensor of a space-time manifold (M,g) by the Bakry-Emery´ Ricci tensor provides us a good apparatus for geometrization of mass.

1 (1) Because of the Einstein equation Ric− Rg = T +E, any attempt on geometrization of mass 2 and electromagnetism then becomes a condition on the Einstein equation and so, ultimately, on the Ricci curvature. Once one has a condition on Ricci curvature, one has a tool to geometrization of mass. Suppose that f is a smooth function on a (M,g), then the m-Bakry-Emery´ Ricci tensor is defined as 1 Ricm = Ric + Hessf − df ⊗ df, f m where, Ric stands for Ricci curvature tensor of (M,g) and m is a non-zero constant that is also ∞ allowed to be infinite, in which case we write Ricf = Ric + Hessf. The Ricci curvature tensor can describes gravity and the m-Bakry-Emery´ Ricci tensor is capable of describing gravity and matter, simultaneously. In fact, the Ricci curvature describes gravity and f can play the role of potential for mass distribution in space-time.

(2) From the point of view of the foundation of , projective structures play an important role; the reason relies on the relations ’free falling (massive) particles-pre-- projective structures’ (see [4]). Let ∇ be the Levi-Civita of a Riemannian manifold (M,g) of dimension n. For a 1-form α define α ∇X Y = ∇X Y − α(X)Y − α(Y )X. The affine connection ∇α is torsion-free, moreover it is projectively equivalent to ∇, meaning that ∇α has the same geodesics, up to re-parametrization. It can be seen that any torsion-free connection projectively equivalent to ∇ is of the form ∇α for some α. df To see the link to Bakry-Emery´ Ricci curvature, let α = , then we recover a Bakry-Emery´ n − 1 Ricci Ricci tensor. As a simple calculation shows that (Proposition 3.3 in [9])

α df ⊗ df Ric∇ = Ric + Hessf + = Ric1−n. g n − 1 f In other words, the Ricci tensor of a projectively equivalent connection is exactly the (1−n)-Bakry- Emery´ Ricci tensor. The Ricci tensor of a projectively equivalent connection can be seen as a condition on Ricci curvature and so can be a tool for geimetrization of mass.

(3) We already know Einstein metrics play a crucial role in general theory of relativity. Hence, the key is provided by generalization of Einstein metrics should probably capable of solve our concern.

2 A triple (M,g,f) (a Riemannian manifold (M,g) with a smooth function f on M) is said to be (m−)quasi-Einstein if it satisfies the equation 1 Ricm = Ric + Hessf − df ⊗ df = λg, f m for some λ ∈ R. The above equation is especially interesting in that when m = ∞ it is exactly the gradient Ricci soliton equation; when m is a positive integer, it corresponds to warped product Einstein metrics (see [3]); when f is constant, it gives the Einstein equation. We call a quasi-Einstein metric trivial when f is constant. Following the terminology of Ricci solitons, a quasi-Einstein metric on a manifold M will be called expanding, steady or shrinking, respectively, if λ < 0, λ = 0 or λ> 0. As we have noticed, the Bakry-Emery´ Ricci tensor is related to notion of quasi Einstein metrics. So, we hope Bakry-Emery´ Ricci provide a good apparatus for geometrization of mass in the next section.

2 New Field Equation

In this section, we consider (M,g) as a space-time manifold whose dimension is n and 2 ≤ n. Each smooth function f on M determines a 1-Bakry-Emery´ Ricci tensor on M. Let M denotes the set of all 1-Bakry-Emery´ Ricci tensor on a manifold M then, M can be written as the set of some pairs (g, f), where g is a semi-Riemannian metric and, f is a smooth function on M. In fact, M can be consider as an open subset of an infinite dimensional . Denote canonical of a meter g on the oriented manifold M by dVg. It is well-known that critical points of Einstein-Hilbert functional

L(g)= RgdVg, ZM are metrics which satisfy the following equation, 1 Ric − Rg =0. 2 The above equation is Einstein field equation in . It motivates to define a new Hilbert- Einstein functional L : M→ R

L(g, f)= R(g,f)dVg, ZM 1 1 where, R(g,f) = tr(Ricf ) is called of Ricf . Equations in which the critical points of this functional satisfy will be called the new field equation. For a symmetric 2-covariant tensor s on M, and a smooth function h set,

g˜(t)= g + ts f˜(t)= f + th

3 For sufficiently small t, (˜g(t), f˜(t)) is a 1-Bakry-Emery´ Ricci tensor on M and is a variation of (g, f). The Bakry-Emery´ Ricci tensor (g, f) is a critical 1-Bakry-Emery´ Ricci tensor for Hilbert action if and only if for any pair (s, h): d d | L g˜(t), f˜(t) = | R dV =0, (2.1) t=0 t=0 Z (˜g(t),f˜(t)) g+ts dt  dt M ˜ where, R(˜g(t),f˜(t)) is the scalar curvature of (˜g(t), f(t)) and

˜ ~ ˜ 2 R(˜g(t),f˜(t)) = Rg˜(t) + △g˜(t)(f(t)) −|∇f(t)| , where, by notation ∇~ we mean the gradient of smooth functions. 2 To find derivation in 2.1, we must compute derivations of Rg˜(t), △g˜(t)(f˜(t)), |∇~ f˜(t)| and dVg+ts for t = 0. In [1], it is shown

′ (Rg˜(t)) (0) = −hs, Rici + div(X), X ∈ X (M), (2.2) 1 (dV )′(0) = hg,sidV . (2.3) g+ts 2 g ˜ ˜ 2 By local computations, we find derivation of △g˜(t)(f(t)) and |∇~ f(t)| at t = 0. By means of a local on M with local frame ∂i, we can write ˜ ˜ d ~ ˜ 2 ij ∂f(t) ∂f(t) ′ ij ∂f ∂f ij ∂h ∂f |t=0|∇f(t)| = g˜ (t) i j (0) = −s i j +2g i j dt ∂x ∂x  ∂x ∂x ∂x ∂x = −hs, df ⊗ dfi +2h∇~ h, ∇~ fi.

Local computation of Laplacian of a smooth function f is as follows. ∂2f ∂f △(f)= gij( − Γk ). ∂xi∂xj ij ∂xk So,

2 ′ ˜ ′ ij ∂ (f + th) ˜k ∂(f + th) [△g˜(t)(f(t))] (0) = g˜ (t)( − Γij(t) ) (0)  ∂xi∂xj ∂xk  ∂2f ∂f ∂2h ∂f ∂h = −sij( − Γk )+ gij( − Ak − Γk ) ∂xi∂xj ij ∂xk ∂xi∂xj ij ∂xk ij ∂xk ∂f = −hs, Hess(f)i + △(f) − gijAk , (2.4) ij ∂xk k ˜k ′ k ij k where, Aij =(Γij) (0). Define a vector field Y by components Y = g Aij. We can write 1 Y k = gijAk = gijgkl(∇ s + ∇ s −∇ s ) ij 2 i jl j il l ij 1 = gkl(gij∇ s + gij∇ s − gij∇ s ) 2 i jl j il l ij 1 1 = gkl(div(s) + div(s) −∇ tr(s)) = div(s)k − (∇~ tr(s))k. 2 l l l 2

4 Consequently, ∂f ∂f 1 ∂f 1 gijAk = div(s)k − (∇~ tr(s))k = div(s)(∇~ f) − h∇~ tr(s), ∇~ fi. ij ∂xk ∂xk 2 ∂xk 2 In the following, whenever it is convenient, we consider s as a (1, 1) . One can easily see ∂f 1 gijAk = div(s(∇~ f)) −hs, Hessfi − h∇~ tr(s), ∇~ fi. ij ∂xk 2 By above computations, we have 1 [△ (f˜(t))]′(0) = △(f) − div(s(∇~ f)) + h∇~ tr(s), ∇~ fi. (2.5) g˜(t) 2 Remind that the integral of divergence of every vector fields on M is zero, consequently integral of Laplacian of any smooth function on M is zero. Also, for any two smooth function f and h we have [8]

h∇~ h, ∇~ fidVg = − △(f)hdVg. ZM ZM Now, we are prepare to find critical 1-Bakry-Emery´ Ricci tensor for Hilbert action. d d 0= | L g˜(t), f˜(t) = | R dV t=0 t=0 Z (˜g(t),f˜(t)) g+ts dt  dt M 1 = R + △(f) −|∇~ f|2 hs,gidV Z g M 2 + −hs, Rici + div(X)+ △(f) − div(s(∇~ f)) Z M 1 + h∇~ tr(s), ∇~ fi + hs, df ⊗ dfi − 2h∇~ h, ∇~ fi dVg 2  1 |∇~ f|2 = h−Ric + Rg + df ⊗ df − g,sidVg ZM 2 2

+ 2△(f)hdVg ZM The above expressions vanish for all pair (g, f) if and only if

1 |∇~ f|2 Ric − Rg = df ⊗ df − g , (2.6) 2 2  △(f)=0. (2.7)

In the case which dimM = 4, taking from both side of 2.6 yields R = |∇~ f|2. The scalar curvature R is related to matter distribution in points of the space-time manifold. Hence, f is related to matter distribution. In fact, equation 2.7 express that div(∇~ f) = 0 and, ∇~ f can be interpreted as current of mass which satisfies conservation law. Also, we can interpret the expression

5 appeared in the right hand side of 2.6 as stress-energy tensor of matter. We need to show that the divergence of |∇~ f|2 T f := df ⊗ df − g, 2 which retrieves conservation law of stress-energy tensor. Theorem 2.1 Suppose that f is a smooth function on a Riemannian manifold (M,g) such that △(f)=0, then the symmetric tensor T f is divergence-free.

n i n Proof 2.2 Let {ei}i=1 is an orthonormal base on M and denote its reciprocal base by {e }i=1. So, we can write n i div(df ⊗ df)(X)= (∇ei df ⊗ df)(e ,X) Xi=1 n i = (∇ei df) ⊗ df + df ⊗ (∇ei df) (e ,X) Xi=1  n i i = (∇ei df)(e )df(X)+ df(e )(∇ei df)(X) Xi=1 = △(f)df(X) + Hessf(∇~ f,X) = Hessf(∇~ f,X).

Also,

2 2 div(|∇~ f| g)(X)= d(|∇~ f| )(X)= Xh∇~ f, ∇~ fi =2h∇X (∇~ f), ∇~ fi

= 2(∇X df)(∇~ f) = 2Hessf(∇~ f,X). The above computations show that div(T f )=0.

In the case which dimM ≥ 3, a simpler form of equation 2.6 is obtained. Theorem 2.3 Suppose that dimM = n ≥ 3, then the critical Bakry-Emery´ tensors of Hilbert action satisfy the following equations.

Ric = df ⊗ df, (2.8) △(f)=0. (2.9)

Proof 2.4 The second equation is the same as 2.7. Let us assume that 2.6 holds, then by taking trace on both sides of this equation, we have n n R − R =(|∇~ f|2 − |∇~ f|2) ⇒ R = |∇~ f|2. 2 2 By replacing the above formula in 2.6, we derive 2.8. Now, let 2.8 holds. By computation traces of two sides of equation 2.8, we find R = |∇~ f|2. Hence, by addition suitable expression to each side of equation 2.8 we obtain 2.6.

6 Our obtained field equations are purely geometric when (g,f,λ) be a 1-quasi Einstein metric for some λ ∈ R.

Theorem 2.5 Suppose that (g, f) is a critic point of Hilbert-Einstein functional then, (g,f,λ) is a 1-quasi Einstein metric if and only if λ = 0 (the quasi-Einstein metric is steady) and ∇~ f is a parallel vector field.

Proof 2.6 Assume (g,f,λ) is 1-quasi Einstein metric on M, then

1 Ricf = Ric + Hessf − df ⊗ df = λg. Hence, the equation 2.8 implies that Hessf = λg. By taking traces on both sides of the above formula and using 2.9, we get

0= λn, so, λ =0 which means the quasi Einstein metric is steady and we have

Hessf =0, which implies that ∇~ f is a parallel vector field on M. Invesely, by definition of steady 1-quasi Einstein metric (g, f) we can write

Ric + Hessf − df ⊗ df =0.

Since, ∇~ f is a parallel vector field, we have Hessf =0 which implies equations 2.8 and 2.9.

Remark: Denote by θ the 1-form df appeared in above theorem. In the Riemannian manifold (M,g) Hessf = ∇df =0, means θ is a parallel 1-form. Moreover, in the generic case (where Ric is non degenerate) by the Weitzenb¨ock formula we have θ = 0. Hence, in this case f is locally constant and because of connectedness of M it will be globally constant. So, generic Riemannian metrics are an obstruction to geometrize matter in general relativity.

3 An example of space-time-mass manifold

In this section we give an example of a manifold which satisfies the equations 2.8 and 2.9. This example is an Einstein-de Sitter model in general relativity. Suppose that (N, g¯) is a Riemannian manifold of dimension n ≥ 2 and, set M = N × (0, ∞). Any tangent vector to M at a point (p, t) is of the form (v,λ) in which v ∈ TpN and λ ∈ R is a scalar. The vector field (0, 1) on M is denoted by ∂t. Considering this vector field as a derivation of ∞ C (M), for a smooth function h(p, t) on M we have ∂t(h)= ∂h/∂t. Denote vector fields on N by X,Y,Z, ··· and consider them as special vector fields on M. Also, denote the second projection map

7 (p, t) 7→ t on M by t. We can interpret t as time. Note that for the 1-form dt we have dt(v,λ)= λ, so dt(∂t) = 1. For some smooth function a : (0, ∞) → R define a metric g on M as follows.

g = e2a(t)g¯ − dt ⊗ dt.

Consider f : N × (0, ∞) → R which depends only on t and denote it by f(t). Consequently, ′ ′ 2 ′ 2 df = f (t)dt, ∇~ f = −f (t)∂t and |∇~ f| = −|f (t) |. Let ∇ and ∇ denote the Levi-Civita connections of N and M, respectively. Routine computations show that

′ ∇X ,Y = ∇X ,Y + a (t)g(X,Y )∂t, ′ ∇X , ∂t = a (t)Y,

∇∂t ∂t =0.

Also, an easy local computation indicates that

△(f)= −na′(t)f ′(t) − f ′′(t). (3.1)

Hence, equation △(f) = 0 implies that |f ′(t)| = ce−na(t), for some constant c. Now, let Ric and Ric denote Ricci curvature tensors of N and M, respectively. Straightforward computations show that

Ric(X,Y )= Ric(X,Y )+(a′′(t)+ na′(t))g(X,Y ),

Ric(X, ∂t)=0, ′′ ′ Ric(∂t, ∂t)= −n(a (t)+ a (t)).

Equation 2.8 for this example, becomes Ric = |f ′(t)|2dt ⊗ dt. Hence, this equation holds if and only if

Ric(X,Y )= −(a′′(t)+ na′(t))g(X,Y ), (3.2) n(a′′(t)+ a′(t)) = |f ′(t)|2 = c2e−2na(t) (3.3)

Left side of 3.2 does not depend on t, so a′′(t)+ na′(t) have to be constant and N is an . In the case a′′(t)+ na′(t) = 0 we find solution a(t)=1/n. ln t and for this solution, equation 3.3 also holds for c = (n − 1)/n. So, for an n-dimensional Riccip flat manifold (N, g¯) the meter g = t2/ng¯ − dt ⊗ dt on M and the function f(t)= (n − 1)/n ln t satisfies field equations 2.8 and 2.9. In this model, as t approaches to zero, universep become smaller and density of matter increases to infinity. Time t = 0 is not in M and this time is the instant of Big-Bang.

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